82.4.12 problem 28-12

Internal problem ID [21830]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-12
Date solved : Thursday, October 02, 2025 at 08:02:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+8 y&=\sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.106 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+8*y(t) = sin(t); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left (138 \cos \left (t \right )^{2}+131 \cos \left (t \right ) \sin \left (t \right )-69\right ) {\mathrm e}^{-2 t}}{65}+\frac {7 \sin \left (t \right )}{65}-\frac {4 \cos \left (t \right )}{65} \]

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 45

ode=D[y[t],{t,2}]+4*D[y[t],t]+8*y[t]==Sin[t]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{65} e^{-2 t} \left (7 e^{2 t} \sin (t)+69 \cos (2 t)+\left (131 \sin (t)-4 e^{2 t}\right ) \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.060 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(8*y(t) - sin(t) + 5*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sin {\left (t \right )}}{3} - \frac {\sqrt {10} \sin {\left (\frac {2 \sqrt {10} t}{5} \right )}}{12} + \cos {\left (\frac {2 \sqrt {10} t}{5} \right )} \]